A328521 -id:A328521

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A354880

Numbers k where A133411(k) is not equal to A328521(k-1).

+20
1

17, 19, 21, 51, 52, 55, 56, 57, 58, 59, 60, 61, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 127, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

A133411(17)=5587021440. (5587021440/19 = A133411(16) = 294053760.) But A328521(16)=2205403200. (2205403200 has 16 prime factors and is A328521(16) even if not 294053760 times an integer [the quotient is 15/2].) Thus, 17 is a term of this sequence.

CROSSREFS

Cf. A133411, A328521.

KEYWORD

nonn

AUTHOR

J. Lowell, Jun 09 2022

STATUS

approved

A133411

Smallest highly composite number of the form k*a(n-1) where k is an integer greater than 1.

+10
5

1, 2, 4, 12, 24, 48, 240, 720, 5040, 10080, 20160, 221760, 665280, 8648640, 17297280, 294053760, 5587021440, 27935107200, 642507465600, 1927522396800, 13492656777600, 26985313555200, 782574093100800, 24259796886124800

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: subsequence of A019505.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..223 (computed from the 10000 term b-file of A002182 prepared from Flammenkamp's data)

EXAMPLE

6 is not in the sequence because 6 is not a multiple of 4, the previous term.

PROG

(PARI)

sublist_of_first_proper_multiple_terms_of(v) = { my(u=v[1], lista=List(u)); for(i=2, #v, if((v[i]>u)&&!(v[i]%u), u = v[i]; listput(lista, u))); Vec(lista); };

v133411 = sublist_of_first_proper_multiple_terms_of(v002182); \\ v002182 contains the terms of A002182.

A133411(n) = v133411[n]; \\ Antti Karttunen, Jan 10 2020

CROSSREFS

Cf. A002182, A019505, A328521, A330744 (primorial deflation).

KEYWORD

nonn

AUTHOR

J. Lowell, Nov 25 2007

EXTENSIONS

a(12)-a(24) from Donovan Johnson, Sep 09 2008

STATUS

approved

A330743

a(n) is the first term k of A329902 for which A056239(k) = n.

+10
4

1, 2, 4, 6, 12, 24, 40, 60, 84, 168, 336, 528, 792, 936, 1872, 2448, 3060, 4560, 4788, 8280, 15456, 23184, 29232, 31248, 62496, 74592, 124320, 137760, 144480, 157920, 315840, 356160, 559680, 623040, 644160, 966240, 1061280, 1124640, 1686960, 1734480, 2049840, 2218320, 2330640, 2499120, 4165200, 4539600, 4726800, 4820400

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Note that in contrast to A330744 this is not monotonic. The first point where a(n) > a(n+1) occurs is at a(120) = 5481774144 > a(121) = 5452302240. See also comment in A328521, whose primorial deflation this sequence is.

a(n-1) differs from A330744(n) at n = 17, 19, 21, 51, 52, 55, 56, 57, 58, 59, 60, 61, ...

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..4257

FORMULA

a(n) = A329902(min{i: A056239(A329902(i))==n}).

a(n) = A329902(A330748(n)).

a(n) = A329900(A328521(n)) = A319626(A328521(n)).

PROG

(PARI) A330743(n) = { for(k=1, oo, if(A056239(A329902(k))==n, return(A329902(k)))); };

(PARI)

v329902 = readvec("a329902.txt"); \\ File for the first 779674 terms of A329902 as prepared by Michael De Vlieger.

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));

A330743list() = { my(m=Map(), lista=List([]), t); for(i=1, #v329902, t = A056239(v329902[i]); if(!mapisdefined(m, t), mapput(m, t, v329902[i]))); for(n=0, oo, if(mapisdefined(m, n, &t), listput(lista, t), return(Vec(lista)))); };

v330743 = A330743list();

A330743(n) = v330743[1+n];

for(n=0, #v330743-1, write("b330743.txt", n, " ", A330743(n)));

CROSSREFS

Primorial deflation of A328521.

Cf. A056239, A319626, A329900, A329902, A330748.

Cf. also A330744.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 13 2020

STATUS

approved

A330748

Index of the smallest element in A002182 that has exactly n prime factors counted with multiplicity.

+10
4

1, 2, 3, 5, 6, 8, 12, 14, 19, 21, 23, 32, 37, 47, 50, 62, 70, 80, 91, 99, 105, 109, 124, 140, 143, 159, 166, 182, 198, 217, 221, 240, 253, 276, 297, 304, 327, 352, 357, 381, 398, 424, 449, 475, 485, 512, 540, 570, 584, 617, 642, 676, 704, 738, 765, 770, 805, 841, 877, 913, 937, 949, 985, 1021, 1058, 1096, 1134, 1169

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..4257 (computed using A. Flammenkamp's 779674-term HCN dataset; terms 0..329 from Antti Karttunen)

FORMULA

a(n) = min{k: A112778(k)=n}.

A002182(a(n)) = A328521(n).

A329902(a(n)) = A330743(n).

PROG

(PARI) A330748(n) = { for(k=1, #v112778, if(v112778[k]==n, return(k))); -(1/0); };

(PARI)

v329902 = readvec("a329902.txt"); \\ File for the first 779674 terms of A329902

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));

A330748list() = { my(m=Map(), lista=List([]), t); for(i=1, #v329902, t = A056239(v329902[i]); if(!mapisdefined(m, t), mapput(m, t, i))); for(n=0, oo, if(mapisdefined(m, n, &t), listput(lista, t), return(Vec(lista)))); };

v330748 = A330748list();

A330748(n) = v330748[1+n];

for(n=0, #v330748-1, write("b330748.txt", n, " ", A330748(n))); \\ Antti Karttunen, Jan 13 2020

CROSSREFS

Cf. A001222, A002182, A112778, A328521, A329902, A330743.

KEYWORD

nonn

AUTHOR

Antti Karttunen, suggested by M. F. Hasler, Jan 10 2020

STATUS

approved

A328522

Largest highly composite number that has n prime factors counted with multiplicity.

+10
0

1, 2, 6, 12, 60, 180, 1260, 2520, 27720, 83160, 1081080, 3603600, 61261200, 698377680, 3491888400, 80313433200, 240940299600, 1686582097200, 48910880818800, 1516237305382800, 3032474610765600, 112201560598327200, 3066842656354276800, 131874234223233902400, 659371171116169512000, 30990445042459967064000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

These numbers are upper bounds on the largest term in A002182 that is not divisible by k for some k.

LINKS

Table of n, a(n) for n=0..25.

MATHEMATICA

With[{s = Take[Import["https://oeis.org/b002182.txt", "Data"][[All, -1]], 240]}, Drop[TakeWhile[#, IntegerQ], -6] &@ Table[s[[Lookup[#, n][[-1]] ]], {n, 0, Max@ Keys@ #}] &@ PositionIndex[Map[PrimeOmega, s]]] (* Michael De Vlieger, Jan 19 2020, using b-file at A002182. Caution: ensure full population of a given value of bigomega by extending scope beyond the desired number of terms. *)

CROSSREFS

Cf. A001222, A002182, A112778, A328521.

KEYWORD

nonn

AUTHOR

David A. Corneth, Jan 04 2020

STATUS

approved

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